Let \(\tilde{U}=[0,1]\times (0,\infty ) \times [0, \infty )\) . Let \(\tilde{b}: \tilde{U}\rightarrow \mathbb {R}^3\)  be the vector field
given by the right-hand sides of (REF ) to ().
Let \(\tilde{\zeta }=\inf \lbrace t>0:\tilde{i}_E(t)=0\rbrace \)  and \(\tau =1-\tilde{s}(\tilde{\zeta })\) . Then it follows from () that
\(\tau \)  is given by the smallest solution in \((\epsilon ,1)\)  of \(F_{\epsilon }(x)=0\) , where \(F_{\epsilon }\)  is defined at (REF ).
(We show in Appendix REF  that such a solution exists.)
We follow Section 4.2 of [1]}, adapted to \(L^2\)  estimates rather than exponential estimates as there presented. For \(\epsilon ^{\prime }>0\) , let
\(\tilde{\zeta }_{\epsilon ^{\prime }}^-&=\inf \lbrace t \ge 0: B(\tilde{x}(t),\epsilon ^{\prime }) \cap \tilde{U}^c \ne \emptyset \rbrace ,\\\tilde{\zeta }_{\epsilon ^{\prime }}^+&=\inf \lbrace t \ge 0: B(\tilde{x}(t),\epsilon ^{\prime }) \subseteq \tilde{U}^c\rbrace ,\) 
